On paracompactness in function spaces with the compact-open topology.

Abstract

Abstract.A k-network (P for a space X is a family of subsets of X such that if CC U, with C compact and U open, then there is a finite union R of members of (P such that CCLRQ U. An Ha-space is a r3-space having a countable ¿-network and an it-space is a Tr space having a <r-locally finite ¿-network. In this paper, it is shown that if X is an No-space and F is a paracompact N-space, then G(X, Y), with the compact-open topology is a paracompact N-space. The result implies that if X is separable metric and Y is metric, then G(X, Y) is paracompact.The purpose of this paper is to show that certain function spaces are paracompact. We shall introduce a class of spaces called N-spaces which share many of the properties of E. , though an N-space need not be Lindelöf. The No-spaces can be characterized as Lindelöf or hereditarily separable N-spaces [7]. Throughout, e(X, Y) will denote the space of continuous functions from X to Y equipped with the compact-open topology. Michael has shown that if X and Y are No-spaces, then so is Q(X, Y). This theorem is generalized in Theorem 1, which states that if X is an N0-space and F is a paracompact N-space, then &(X, Y) is a paracompact N-space. It is unknown whether the preceding statement is true when "paracompact" is deleted. Definition 1. A k-network (P for a space X is a family of subsets of X such that if CC U, with C compact and U open in X, then there is a finite union R of members of (P such that CERE U. A network (P for a space X is a family of subsets of X such that if xEU, with U open, then there is a P£(P such that xEPEU.In the preceding definition of fe-network, if the finite union R is replaced by a member P£(P, then (P is called a pseudobase for X. Regular Pi-spaces with countable pseudobases (i.e., N0-spaces) and spaces with c-locally finite networks (i.e., <r-spaces) have been investigated by E. Michael and A. Okuyama in [3] and [S], respectively. Though a ^-network may not be a pseudobase, a space with a countable ^-network also has a countable pseudobase.

Products of 𝑘-spaces and spaces of countable tightness

“…As applications of results in §1, we shall consider the products of ¿-spaces and spaces of countable tightness in more special cases. Definition 2.1 [8,16]. A collection 9 of (not necessarily open) subsets of a space A is a k-network for A if, whenever CEU with C compact and U open, then C C U f C U for some finite subcollection § of 9.…”

On defining a space by a weak base

“…A collection & of subsets of X is a cs-network for X if whenever U is an open set which contains a convergent sequence {z n } and also contains the limit z of the convergent sequence {z n }> then {z, z n , z n+1 , " }cPc U for some n and for some P in &*. A space X is an ^-space if it is regular and has a ^-locally finite ^-network [28]. A space X is an # Q -space if it is regular and has a countablê -network [23].…”

k-systems, k-networks and k-covers

“…[10]) if X is determined by P and each element of P is compact (resp. metric and compact) in X. P is called a k-network for X if, whenever K ⊂ U with K compact and U open in X, then K ⊂ P ⊂ U for some finite P ⊂ P [14]. P is called a compact (resp.…”